Left Termination of the query pattern p_in_1(g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 PiDP
    ↳7 PiDPToQDPProof (⇐)
    ↳8 QDP
    ↳9 QDPOrderProof (⇔)
    ↳10 QDP
    ↳11 QDPOrderProof (⇔)
    ↳12 QDP
    ↳13 DependencyGraphProof (⇔)
    ↳14 QDP
    ↳15 UsableRulesProof (⇔)
    ↳16 QDP
    ↳17 QReductionProof (⇔)
    ↳18 QDP
    ↳19 UsableRulesReductionPairsProof (⇔)
    ↳20 QDP
    ↳21 PisEmptyProof (⇔)
    ↳22 TRUE
  ↳23 PrologToPiTRSProof (⇐)
    ↳24 PiTRS
    ↳25 DependencyPairsProof (⇔)
    ↳26 PiDP
    ↳27 DependencyGraphProof (⇔)
    ↳28 PiDP
    ↳29 PiDPToQDPProof (⇔)
    ↳30 QDP
    ↳31 QDPOrderProof (⇔)
    ↳32 QDP

(0) Obligation:

Clauses:

p(.(X, [])).
p(.(s(s(X)), .(Y, Xs))) :- ','(p(.(X, .(Y, Xs))), p(.(s(s(s(s(Y)))), Xs))).
p(.(0, Xs)) :- p(Xs).

Queries:

p(g).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
p_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g(.(X, []))
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(Xs, p_in_g(Xs))
U3_g(Xs, p_out_g(Xs)) → p_out_g(.(0, Xs))
U1_g(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_g(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(X, Y, Xs, p_out_g(.(s(s(s(s(Y)))), Xs))) → p_out_g(.(s(s(X)), .(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
p_out_g(x1)  =  p_out_g
s(x1)  =  s(x1)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
0  =  0
U3_g(x1, x2)  =  U3_g(x2)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g(.(X, []))
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(Xs, p_in_g(Xs))
U3_g(Xs, p_out_g(Xs)) → p_out_g(.(0, Xs))
U1_g(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_g(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(X, Y, Xs, p_out_g(.(s(s(s(s(Y)))), Xs))) → p_out_g(.(s(s(X)), .(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
p_out_g(x1)  =  p_out_g
s(x1)  =  s(x1)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
0  =  0
U3_g(x1, x2)  =  U3_g(x2)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

P_IN_G(.(s(s(X)), .(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))
P_IN_G(.(0, Xs)) → U3_G(Xs, p_in_g(Xs))
P_IN_G(.(0, Xs)) → P_IN_G(Xs)
U1_G(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_G(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U1_G(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → P_IN_G(.(s(s(s(s(Y)))), Xs))

The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g(.(X, []))
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(Xs, p_in_g(Xs))
U3_g(Xs, p_out_g(Xs)) → p_out_g(.(0, Xs))
U1_g(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_g(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(X, Y, Xs, p_out_g(.(s(s(s(s(Y)))), Xs))) → p_out_g(.(s(s(X)), .(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
p_out_g(x1)  =  p_out_g
s(x1)  =  s(x1)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
0  =  0
U3_g(x1, x2)  =  U3_g(x2)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x2, x3, x4)
U3_G(x1, x2)  =  U3_G(x2)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)

We have to consider all (P,R,Pi)-chains

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P_IN_G(.(s(s(X)), .(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))
P_IN_G(.(0, Xs)) → U3_G(Xs, p_in_g(Xs))
P_IN_G(.(0, Xs)) → P_IN_G(Xs)
U1_G(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_G(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U1_G(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → P_IN_G(.(s(s(s(s(Y)))), Xs))

The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g(.(X, []))
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(Xs, p_in_g(Xs))
U3_g(Xs, p_out_g(Xs)) → p_out_g(.(0, Xs))
U1_g(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_g(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(X, Y, Xs, p_out_g(.(s(s(s(s(Y)))), Xs))) → p_out_g(.(s(s(X)), .(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
p_out_g(x1)  =  p_out_g
s(x1)  =  s(x1)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
0  =  0
U3_g(x1, x2)  =  U3_g(x2)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x2, x3, x4)
U3_G(x1, x2)  =  U3_G(x2)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)

We have to consider all (P,R,Pi)-chains

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U1_G(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → P_IN_G(.(s(s(s(s(Y)))), Xs))
P_IN_G(.(s(s(X)), .(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))
P_IN_G(.(0, Xs)) → P_IN_G(Xs)

The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g(.(X, []))
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(Xs, p_in_g(Xs))
U3_g(Xs, p_out_g(Xs)) → p_out_g(.(0, Xs))
U1_g(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_g(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(X, Y, Xs, p_out_g(.(s(s(s(s(Y)))), Xs))) → p_out_g(.(s(s(X)), .(Y, Xs)))

The argument filtering Pi contains the following mapping:
p_in_g(x1)  =  p_in_g(x1)
.(x1, x2)  =  .(x1, x2)
[]  =  []
p_out_g(x1)  =  p_out_g
s(x1)  =  s(x1)
U1_g(x1, x2, x3, x4)  =  U1_g(x2, x3, x4)
0  =  0
U3_g(x1, x2)  =  U3_g(x2)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
P_IN_G(x1)  =  P_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x2, x3, x4)

We have to consider all (P,R,Pi)-chains

(7) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(8) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U1_G(Y, Xs, p_out_g) → P_IN_G(.(s(s(s(s(Y)))), Xs))
P_IN_G(.(s(s(X)), .(Y, Xs))) → U1_G(Y, Xs, p_in_g(.(X, .(Y, Xs))))
P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))
P_IN_G(.(0, Xs)) → P_IN_G(Xs)

The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(p_in_g(Xs))
U3_g(p_out_g) → p_out_g
U1_g(Y, Xs, p_out_g) → U2_g(p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(p_out_g) → p_out_g

The set Q consists of the following terms:

p_in_g(x0)
U3_g(x0)
U1_g(x0, x1, x2)
U2_g(x0)

We have to consider all (P,Q,R)-chains.

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


P_IN_G(.(0, Xs)) → P_IN_G(Xs)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = x1 + x2   
POL(0) = 1   
POL(P_IN_G(x1)) = x1   
POL(U1_G(x1, x2, x3)) = x1 + x2   
POL(U1_g(x1, x2, x3)) = 0   
POL(U2_g(x1)) = 0   
POL(U3_g(x1)) = 0   
POL([]) = 0   
POL(p_in_g(x1)) = 0   
POL(p_out_g) = 0   
POL(s(x1)) = x1   

The following usable rules [FROCOS05] were oriented: none

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U1_G(Y, Xs, p_out_g) → P_IN_G(.(s(s(s(s(Y)))), Xs))
P_IN_G(.(s(s(X)), .(Y, Xs))) → U1_G(Y, Xs, p_in_g(.(X, .(Y, Xs))))
P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))

The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(p_in_g(Xs))
U3_g(p_out_g) → p_out_g
U1_g(Y, Xs, p_out_g) → U2_g(p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(p_out_g) → p_out_g

The set Q consists of the following terms:

p_in_g(x0)
U3_g(x0)
U1_g(x0, x1, x2)
U2_g(x0)

We have to consider all (P,Q,R)-chains.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


P_IN_G(.(s(s(X)), .(Y, Xs))) → U1_G(Y, Xs, p_in_g(.(X, .(Y, Xs))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 1 + x2   
POL(0) = 0   
POL(P_IN_G(x1)) = x1   
POL(U1_G(x1, x2, x3)) = 1 + x2   
POL(U1_g(x1, x2, x3)) = 0   
POL(U2_g(x1)) = 0   
POL(U3_g(x1)) = 0   
POL([]) = 0   
POL(p_in_g(x1)) = 0   
POL(p_out_g) = 0   
POL(s(x1)) = 0   

The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U1_G(Y, Xs, p_out_g) → P_IN_G(.(s(s(s(s(Y)))), Xs))
P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))

The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(p_in_g(Xs))
U3_g(p_out_g) → p_out_g
U1_g(Y, Xs, p_out_g) → U2_g(p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(p_out_g) → p_out_g

The set Q consists of the following terms:

p_in_g(x0)
U3_g(x0)
U1_g(x0, x1, x2)
U2_g(x0)

We have to consider all (P,Q,R)-chains.

(13) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(14) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))

The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(p_in_g(Xs))
U3_g(p_out_g) → p_out_g
U1_g(Y, Xs, p_out_g) → U2_g(p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(p_out_g) → p_out_g

The set Q consists of the following terms:

p_in_g(x0)
U3_g(x0)
U1_g(x0, x1, x2)
U2_g(x0)

We have to consider all (P,Q,R)-chains.

(15) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(16) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))

R is empty.
The set Q consists of the following terms:

p_in_g(x0)
U3_g(x0)
U1_g(x0, x1, x2)
U2_g(x0)

We have to consider all (P,Q,R)-chains.

(17) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

p_in_g(x0)
U3_g(x0)
U1_g(x0, x1, x2)
U2_g(x0)

(18) Obligation:

Q DP problem:
The TRS P consists of the following rules:

P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(19) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))
No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(.(x1, x2)) = 2·x1 + x2   
POL(P_IN_G(x1)) = 2·x1   
POL(s(x1)) = 2·x1   

(20) Obligation:

Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(21) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(22) TRUE

(23) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
p_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g(.(X, []))
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(Xs, p_in_g(Xs))
U3_g(Xs, p_out_g(Xs)) → p_out_g(.(0, Xs))
U1_g(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_g(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(X, Y, Xs, p_out_g(.(s(s(s(s(Y)))), Xs))) → p_out_g(.(s(s(X)), .(Y, Xs)))

Pi is empty.

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(24) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g(.(X, []))
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(Xs, p_in_g(Xs))
U3_g(Xs, p_out_g(Xs)) → p_out_g(.(0, Xs))
U1_g(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_g(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(X, Y, Xs, p_out_g(.(s(s(s(s(Y)))), Xs))) → p_out_g(.(s(s(X)), .(Y, Xs)))

Pi is empty.

(25) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

P_IN_G(.(s(s(X)), .(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))
P_IN_G(.(0, Xs)) → U3_G(Xs, p_in_g(Xs))
P_IN_G(.(0, Xs)) → P_IN_G(Xs)
U1_G(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_G(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U1_G(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → P_IN_G(.(s(s(s(s(Y)))), Xs))

The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g(.(X, []))
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(Xs, p_in_g(Xs))
U3_g(Xs, p_out_g(Xs)) → p_out_g(.(0, Xs))
U1_g(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_g(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(X, Y, Xs, p_out_g(.(s(s(s(s(Y)))), Xs))) → p_out_g(.(s(s(X)), .(Y, Xs)))

Pi is empty.
We have to consider all (P,R,Pi)-chains

(26) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

P_IN_G(.(s(s(X)), .(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))
P_IN_G(.(0, Xs)) → U3_G(Xs, p_in_g(Xs))
P_IN_G(.(0, Xs)) → P_IN_G(Xs)
U1_G(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_G(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U1_G(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → P_IN_G(.(s(s(s(s(Y)))), Xs))

The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g(.(X, []))
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(Xs, p_in_g(Xs))
U3_g(Xs, p_out_g(Xs)) → p_out_g(.(0, Xs))
U1_g(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_g(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(X, Y, Xs, p_out_g(.(s(s(s(s(Y)))), Xs))) → p_out_g(.(s(s(X)), .(Y, Xs)))

Pi is empty.
We have to consider all (P,R,Pi)-chains

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(28) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

U1_G(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → P_IN_G(.(s(s(s(s(Y)))), Xs))
P_IN_G(.(s(s(X)), .(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))
P_IN_G(.(0, Xs)) → P_IN_G(Xs)

The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g(.(X, []))
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(Xs, p_in_g(Xs))
U3_g(Xs, p_out_g(Xs)) → p_out_g(.(0, Xs))
U1_g(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_g(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(X, Y, Xs, p_out_g(.(s(s(s(s(Y)))), Xs))) → p_out_g(.(s(s(X)), .(Y, Xs)))

Pi is empty.
We have to consider all (P,R,Pi)-chains

(29) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(30) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U1_G(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → P_IN_G(.(s(s(s(s(Y)))), Xs))
P_IN_G(.(s(s(X)), .(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))
P_IN_G(.(0, Xs)) → P_IN_G(Xs)

The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g(.(X, []))
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(Xs, p_in_g(Xs))
U3_g(Xs, p_out_g(Xs)) → p_out_g(.(0, Xs))
U1_g(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_g(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(X, Y, Xs, p_out_g(.(s(s(s(s(Y)))), Xs))) → p_out_g(.(s(s(X)), .(Y, Xs)))

The set Q consists of the following terms:

p_in_g(x0)
U3_g(x0, x1)
U1_g(x0, x1, x2, x3)
U2_g(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


P_IN_G(.(0, Xs)) → P_IN_G(Xs)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x1, x2)) = x1 + x2   
POL(0) = 1   
POL(P_IN_G(x1)) = x1   
POL(U1_G(x1, x2, x3, x4)) = x2 + x3   
POL(U1_g(x1, x2, x3, x4)) = x1 + x2 + x3   
POL(U2_g(x1, x2, x3, x4)) = x1   
POL(U3_g(x1, x2)) = x2   
POL([]) = 0   
POL(p_in_g(x1)) = x1   
POL(p_out_g(x1)) = 0   
POL(s(x1)) = x1   

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

Q DP problem:
The TRS P consists of the following rules:

U1_G(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → P_IN_G(.(s(s(s(s(Y)))), Xs))
P_IN_G(.(s(s(X)), .(Y, Xs))) → U1_G(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
P_IN_G(.(s(s(X)), .(Y, Xs))) → P_IN_G(.(X, .(Y, Xs)))

The TRS R consists of the following rules:

p_in_g(.(X, [])) → p_out_g(.(X, []))
p_in_g(.(s(s(X)), .(Y, Xs))) → U1_g(X, Y, Xs, p_in_g(.(X, .(Y, Xs))))
p_in_g(.(0, Xs)) → U3_g(Xs, p_in_g(Xs))
U3_g(Xs, p_out_g(Xs)) → p_out_g(.(0, Xs))
U1_g(X, Y, Xs, p_out_g(.(X, .(Y, Xs)))) → U2_g(X, Y, Xs, p_in_g(.(s(s(s(s(Y)))), Xs)))
U2_g(X, Y, Xs, p_out_g(.(s(s(s(s(Y)))), Xs))) → p_out_g(.(s(s(X)), .(Y, Xs)))

The set Q consists of the following terms:

p_in_g(x0)
U3_g(x0, x1)
U1_g(x0, x1, x2, x3)
U2_g(x0, x1, x2, x3)

We have to consider all (P,Q,R)-chains.